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Properties

Label 5915.2.a.bh
Level $5915$
Weight $2$
Character orbit 5915.a
Self dual yes
Analytic conductor $47.232$
Analytic rank $1$
Dimension $15$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5915,2,Mod(1,5915)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5915, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5915.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5915 = 5 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5915.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(47.2315127956\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 2 x^{14} - 24 x^{13} + 49 x^{12} + 220 x^{11} - 458 x^{10} - 958 x^{9} + 2032 x^{8} + 2001 x^{7} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{14} q^{3} + (\beta_{11} - \beta_{10} + 2) q^{4} - q^{5} + (2 \beta_{10} + \beta_{8} - \beta_{5} + \cdots - 1) q^{6}+ \cdots + (\beta_{12} - 2 \beta_{10} - \beta_{4} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{14} q^{3} + (\beta_{11} - \beta_{10} + 2) q^{4} - q^{5} + (2 \beta_{10} + \beta_{8} - \beta_{5} + \cdots - 1) q^{6}+ \cdots + ( - 7 \beta_{14} + \beta_{13} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 2 q^{2} - 6 q^{3} + 22 q^{4} - 15 q^{5} - 10 q^{6} + 15 q^{7} + 3 q^{8} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 2 q^{2} - 6 q^{3} + 22 q^{4} - 15 q^{5} - 10 q^{6} + 15 q^{7} + 3 q^{8} + 17 q^{9} + 2 q^{10} - 4 q^{11} - 34 q^{12} - 2 q^{14} + 6 q^{15} + 28 q^{16} - 9 q^{17} + 26 q^{18} - 28 q^{19} - 22 q^{20} - 6 q^{21} - 18 q^{22} + 8 q^{23} - 39 q^{24} + 15 q^{25} - 15 q^{27} + 22 q^{28} + 21 q^{29} + 10 q^{30} - 18 q^{31} + 34 q^{32} - 48 q^{33} - 55 q^{34} - 15 q^{35} + 50 q^{36} - 36 q^{37} - 49 q^{38} - 3 q^{40} - 8 q^{41} - 10 q^{42} - 18 q^{43} - 13 q^{44} - 17 q^{45} + 8 q^{46} - 25 q^{47} - 65 q^{48} + 15 q^{49} - 2 q^{50} + 30 q^{51} - 14 q^{53} - 52 q^{54} + 4 q^{55} + 3 q^{56} - 6 q^{57} - 10 q^{58} + 21 q^{59} + 34 q^{60} - 8 q^{61} + 30 q^{62} + 17 q^{63} - 9 q^{64} + 56 q^{66} - 42 q^{67} - 19 q^{68} + 52 q^{69} + 2 q^{70} + 21 q^{71} - 13 q^{72} - 23 q^{73} + 34 q^{74} - 6 q^{75} - 72 q^{76} - 4 q^{77} - 28 q^{80} + 27 q^{81} - 7 q^{82} - 32 q^{83} - 34 q^{84} + 9 q^{85} - 66 q^{86} - 5 q^{87} - 15 q^{88} - 34 q^{89} - 26 q^{90} + 70 q^{92} + 33 q^{93} - 11 q^{94} + 28 q^{95} - 43 q^{96} - 93 q^{97} - 2 q^{98} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 2 x^{14} - 24 x^{13} + 49 x^{12} + 220 x^{11} - 458 x^{10} - 958 x^{9} + 2032 x^{8} + 2001 x^{7} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3703465 \nu^{14} + 72201658 \nu^{13} + 155134301 \nu^{12} - 1633325197 \nu^{11} + \cdots + 3786451390 ) / 2060341543 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2423975 \nu^{14} + 17384157 \nu^{13} - 56107402 \nu^{12} - 405165440 \nu^{11} + 526580000 \nu^{10} + \cdots - 34890278 ) / 158487811 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 45847381 \nu^{14} - 124730677 \nu^{13} - 1107544176 \nu^{12} + 3035318882 \nu^{11} + \cdots - 431182670 ) / 2060341543 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 47615803 \nu^{14} - 30221533 \nu^{13} - 1179599840 \nu^{12} + 697480605 \nu^{11} + \cdots - 1810271521 ) / 2060341543 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 51548800 \nu^{14} - 64599343 \nu^{13} - 1200791877 \nu^{12} + 1765040913 \nu^{11} + \cdots - 3991994017 ) / 2060341543 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 65272133 \nu^{14} - 298686144 \nu^{13} - 1597163528 \nu^{12} + 7144021208 \nu^{11} + \cdots + 4510189140 ) / 2060341543 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 69877721 \nu^{14} + 172465374 \nu^{13} + 1722292129 \nu^{12} - 4111087057 \nu^{11} + \cdots - 470557154 ) / 2060341543 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 74141806 \nu^{14} + 37474133 \nu^{13} - 1736662701 \nu^{12} - 646198611 \nu^{11} + \cdots - 150290362 ) / 2060341543 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 104442981 \nu^{14} - 101708241 \nu^{13} - 2461950379 \nu^{12} + 2592246155 \nu^{11} + \cdots + 2600730850 ) / 2060341543 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 104442981 \nu^{14} - 101708241 \nu^{13} - 2461950379 \nu^{12} + 2592246155 \nu^{11} + \cdots - 5640635322 ) / 2060341543 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 106378024 \nu^{14} + 268419043 \nu^{13} + 2545029016 \nu^{12} - 6563409629 \nu^{11} + \cdots + 1866973232 ) / 2060341543 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 107177721 \nu^{14} + 44681165 \nu^{13} - 2525459914 \nu^{12} - 711785643 \nu^{11} + \cdots - 104442981 ) / 2060341543 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 190373150 \nu^{14} + 208291341 \nu^{13} + 4540623469 \nu^{12} - 5314200858 \nu^{11} + \cdots + 1552303867 ) / 2060341543 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{10} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} + \beta_{10} - \beta_{5} + \beta_{4} - \beta_{2} + 5\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} + 6\beta_{11} - 7\beta_{10} + 2\beta_{9} + \beta_{7} + \beta_{5} - \beta_{4} + \beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{14} + 9 \beta_{12} - \beta_{11} + 10 \beta_{10} - \beta_{9} + \beta_{6} - 8 \beta_{5} + 12 \beta_{4} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 12 \beta_{13} - 2 \beta_{12} + 37 \beta_{11} - 47 \beta_{10} + 22 \beta_{9} + 2 \beta_{8} + \cdots + 134 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7 \beta_{14} + \beta_{13} + 69 \beta_{12} - 15 \beta_{11} + 80 \beta_{10} - 17 \beta_{9} - 2 \beta_{7} + \cdots - 112 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{14} - 108 \beta_{13} - 34 \beta_{12} + 236 \beta_{11} - 319 \beta_{10} + 190 \beta_{9} + \cdots + 864 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 31 \beta_{14} + 25 \beta_{13} + 509 \beta_{12} - 157 \beta_{11} + 605 \beta_{10} - 201 \beta_{9} + \cdots - 949 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 21 \beta_{14} - 882 \beta_{13} - 391 \beta_{12} + 1549 \beta_{11} - 2200 \beta_{10} + 1520 \beta_{9} + \cdots + 5778 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 56 \beta_{14} + 370 \beta_{13} + 3721 \beta_{12} - 1432 \beta_{11} + 4492 \beta_{10} - 2019 \beta_{9} + \cdots - 7679 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 285 \beta_{14} - 6916 \beta_{13} - 3828 \beta_{12} + 10421 \beta_{11} - 15377 \beta_{10} + 11796 \beta_{9} + \cdots + 39635 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 763 \beta_{14} + 4333 \beta_{13} + 27210 \beta_{12} - 12213 \beta_{11} + 33143 \beta_{10} + \cdots - 60641 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 3189 \beta_{14} - 53223 \beta_{13} - 34440 \beta_{12} + 71623 \beta_{11} - 108628 \beta_{10} + \cdots + 277077 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.56690
2.42802
2.34918
2.07184
1.78359
1.41971
0.529216
0.0443209
−0.0269045
−0.588790
−1.29472
−1.66049
−2.35871
−2.50716
−2.75600
−2.56690 −2.71021 4.58896 −1.00000 6.95682 1.00000 −6.64558 4.34522 2.56690
1.2 −2.42802 2.49683 3.89529 −1.00000 −6.06235 1.00000 −4.60180 3.23415 2.42802
1.3 −2.34918 −0.724389 3.51863 −1.00000 1.70172 1.00000 −3.56753 −2.47526 2.34918
1.4 −2.07184 1.05752 2.29252 −1.00000 −2.19101 1.00000 −0.606057 −1.88165 2.07184
1.5 −1.78359 −1.67105 1.18119 −1.00000 2.98047 1.00000 1.46042 −0.207590 1.78359
1.6 −1.41971 −0.501451 0.0155700 −1.00000 0.711913 1.00000 2.81731 −2.74855 1.41971
1.7 −0.529216 0.115418 −1.71993 −1.00000 −0.0610809 1.00000 1.96865 −2.98668 0.529216
1.8 −0.0443209 2.91481 −1.99804 −1.00000 −0.129187 1.00000 0.177197 5.49611 0.0443209
1.9 0.0269045 −0.667174 −1.99928 −1.00000 −0.0179500 1.00000 −0.107599 −2.55488 −0.0269045
1.10 0.588790 −2.94947 −1.65333 −1.00000 −1.73662 1.00000 −2.15104 5.69937 −0.588790
1.11 1.29472 1.56800 −0.323711 −1.00000 2.03012 1.00000 −3.00855 −0.541371 −1.29472
1.12 1.66049 2.16057 0.757214 −1.00000 3.58759 1.00000 −2.06363 1.66804 −1.66049
1.13 2.35871 −2.29186 3.56352 −1.00000 −5.40582 1.00000 3.68788 2.25260 −2.35871
1.14 2.50716 −3.44574 4.28585 −1.00000 −8.63901 1.00000 5.73099 8.87310 −2.50716
1.15 2.75600 −1.35181 5.59555 −1.00000 −3.72559 1.00000 9.90934 −1.17261 −2.75600
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(-1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5915.2.a.bh ✓ 15
13.b even 2 1 5915.2.a.bk yes 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5915.2.a.bh ✓ 15 1.a even 1 1 trivial
5915.2.a.bk yes 15 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5915))\):

\( T_{2}^{15} + 2 T_{2}^{14} - 24 T_{2}^{13} - 49 T_{2}^{12} + 220 T_{2}^{11} + 458 T_{2}^{10} - 958 T_{2}^{9} + \cdots - 1 \) Copy content Toggle raw display
\( T_{3}^{15} + 6 T_{3}^{14} - 13 T_{3}^{13} - 133 T_{3}^{12} - 34 T_{3}^{11} + 1050 T_{3}^{10} + \cdots + 104 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + 2 T^{14} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{15} + 6 T^{14} + \cdots + 104 \) Copy content Toggle raw display
$5$ \( (T + 1)^{15} \) Copy content Toggle raw display
$7$ \( (T - 1)^{15} \) Copy content Toggle raw display
$11$ \( T^{15} + 4 T^{14} + \cdots - 1063091 \) Copy content Toggle raw display
$13$ \( T^{15} \) Copy content Toggle raw display
$17$ \( T^{15} + 9 T^{14} + \cdots - 56717928 \) Copy content Toggle raw display
$19$ \( T^{15} + 28 T^{14} + \cdots + 14466808 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 1636013119 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 12355446839 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 41177808568 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 65167361741 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 54307238336 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 3873718003 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 17662380632 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 2932437846353 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 195127443496 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 9664240664 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 11264382998413 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 267789955007 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 12870643064 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 5980014391 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 668189248 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 78983786110648 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 34044615592 \) Copy content Toggle raw display
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